/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 32 Write the first three terms in e... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 14: Problem 32

Write the first three terms in each binomial expansion, expressing the result in simplified form. $$(x+3)^{8}$$

Short Answer

Expert verified
The first three terms in the expansion of \( (x+3)^{8} \) are \( 6561, 17496x, 20412x^{2} \).

Step by step solution

01

Understanding the Binomial Theorem

The binomial theorem describes the expansion of powers of a binomial. According to this theorem, \( (x+y)^n = \sum_{k=0}^{n} \binom{n}{k} (x)^k (y)^{n-k} \). It means that to simplify \( (x+3)^{8} \) into its first three terms, these binomial coefficients need to be determined.
02

Calculate the first term

The first term in the expansion is when \( k=0 \) in the binomial theorem. Substituting \( k=0 \), \( n=8 \), \( x=x \) and \( y=3 \) into the binomial theorem gives the first term. Hence, first term = \(\binom{8}{0} (x)^0 (3)^{8-0} = \binom{8}{0} * 1 * 3^{8} = 1*3^{8} = 6561 \)
03

Calculate the second term

Next, the second term in the expansion can be calculated by substituting \( k=1 \) into the binomial theorem. Hence, second term = \(\binom{8}{1} (x)^1 (3)^{8-1} = 8*x*(3)^7 = 8x*2187 = 17496x \)
04

Calculate the third term

Finally, the third term in the expansion can be calculated by substituting \( k=2 \) into the binomial theorem. Hence, third term = \(\binom{8}{2} (x)^2 (3)^{8-2} = 28 * x^{2} * 729 = 20412x^{2} \)

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A deposit of 6000 dollars is made in an account that earns \(6 \%\) interest compounded quarterly. The balance in the account after \(n\) quarters is given by the sequence $$a_{n}=6000\left(1+\frac{0.06}{4}\right)^{n}, \quad n=1,2,3, \ldots$$ Find the balance in the account after five years. Round to the nearest cent.

Subtract: \(\frac{x}{x+3}-\frac{x+1}{2 x^{2}-2 x-24}\). (Section 7.4, Example 7)

Find a general term, \(a_{n},\) for each sequence. More than one answer may be possible. $$1 \cdot 3,2 \cdot 4,3 \cdot 5,4 \cdot 6, \dots$$

Solve for \(P: A=\frac{P t}{P+t}\)

What is the difference between a geometric sequence and an infinite geometric series?
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.